3.18.27 \(\int \frac {(1+x^5) \sqrt [3]{1+x^3+x^5} (-3+2 x^5)}{x^2 (2-2 x^3+4 x^5-x^6-2 x^8+2 x^{10})} \, dx\)

Optimal. Leaf size=116 \[ \frac {3 \sqrt [3]{x^5+x^3+1}}{2 x}-\frac {1}{4} \text {RootSum}\left [2 \text {$\#$1}^6-6 \text {$\#$1}^3+3\& ,\frac {-4 \text {$\#$1}^3 \log \left (\sqrt [3]{x^5+x^3+1}-\text {$\#$1} x\right )+4 \text {$\#$1}^3 \log (x)+3 \log \left (\sqrt [3]{x^5+x^3+1}-\text {$\#$1} x\right )-3 \log (x)}{2 \text {$\#$1}^5-3 \text {$\#$1}^2}\& \right ] \]

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Rubi [F]  time = 2.84, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \begin {gather*} \int \frac {\left (1+x^5\right ) \sqrt [3]{1+x^3+x^5} \left (-3+2 x^5\right )}{x^2 \left (2-2 x^3+4 x^5-x^6-2 x^8+2 x^{10}\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[((1 + x^5)*(1 + x^3 + x^5)^(1/3)*(-3 + 2*x^5))/(x^2*(2 - 2*x^3 + 4*x^5 - x^6 - 2*x^8 + 2*x^10)),x]

[Out]

(-3*Defer[Int][(1 + x^3 + x^5)^(1/3)/x^2, x])/2 - 3*Defer[Int][(x*(1 + x^3 + x^5)^(1/3))/(2 - 2*x^3 + 4*x^5 -
x^6 - 2*x^8 + 2*x^10), x] + 5*Defer[Int][(x^3*(1 + x^3 + x^5)^(1/3))/(2 - 2*x^3 + 4*x^5 - x^6 - 2*x^8 + 2*x^10
), x] - (3*Defer[Int][(x^4*(1 + x^3 + x^5)^(1/3))/(2 - 2*x^3 + 4*x^5 - x^6 - 2*x^8 + 2*x^10), x])/2 - 3*Defer[
Int][(x^6*(1 + x^3 + x^5)^(1/3))/(2 - 2*x^3 + 4*x^5 - x^6 - 2*x^8 + 2*x^10), x] + 5*Defer[Int][(x^8*(1 + x^3 +
x^5)^(1/3))/(2 - 2*x^3 + 4*x^5 - x^6 - 2*x^8 + 2*x^10), x]

Rubi steps

\begin {align*} \int \frac {\left (1+x^5\right ) \sqrt [3]{1+x^3+x^5} \left (-3+2 x^5\right )}{x^2 \left (2-2 x^3+4 x^5-x^6-2 x^8+2 x^{10}\right )} \, dx &=\int \left (-\frac {3 \sqrt [3]{1+x^3+x^5}}{2 x^2}+\frac {x \sqrt [3]{1+x^3+x^5} \left (-6+10 x^2-3 x^3-6 x^5+10 x^7\right )}{2 \left (2-2 x^3+4 x^5-x^6-2 x^8+2 x^{10}\right )}\right ) \, dx\\ &=\frac {1}{2} \int \frac {x \sqrt [3]{1+x^3+x^5} \left (-6+10 x^2-3 x^3-6 x^5+10 x^7\right )}{2-2 x^3+4 x^5-x^6-2 x^8+2 x^{10}} \, dx-\frac {3}{2} \int \frac {\sqrt [3]{1+x^3+x^5}}{x^2} \, dx\\ &=\frac {1}{2} \int \left (-\frac {6 x \sqrt [3]{1+x^3+x^5}}{2-2 x^3+4 x^5-x^6-2 x^8+2 x^{10}}+\frac {10 x^3 \sqrt [3]{1+x^3+x^5}}{2-2 x^3+4 x^5-x^6-2 x^8+2 x^{10}}-\frac {3 x^4 \sqrt [3]{1+x^3+x^5}}{2-2 x^3+4 x^5-x^6-2 x^8+2 x^{10}}-\frac {6 x^6 \sqrt [3]{1+x^3+x^5}}{2-2 x^3+4 x^5-x^6-2 x^8+2 x^{10}}+\frac {10 x^8 \sqrt [3]{1+x^3+x^5}}{2-2 x^3+4 x^5-x^6-2 x^8+2 x^{10}}\right ) \, dx-\frac {3}{2} \int \frac {\sqrt [3]{1+x^3+x^5}}{x^2} \, dx\\ &=-\left (\frac {3}{2} \int \frac {\sqrt [3]{1+x^3+x^5}}{x^2} \, dx\right )-\frac {3}{2} \int \frac {x^4 \sqrt [3]{1+x^3+x^5}}{2-2 x^3+4 x^5-x^6-2 x^8+2 x^{10}} \, dx-3 \int \frac {x \sqrt [3]{1+x^3+x^5}}{2-2 x^3+4 x^5-x^6-2 x^8+2 x^{10}} \, dx-3 \int \frac {x^6 \sqrt [3]{1+x^3+x^5}}{2-2 x^3+4 x^5-x^6-2 x^8+2 x^{10}} \, dx+5 \int \frac {x^3 \sqrt [3]{1+x^3+x^5}}{2-2 x^3+4 x^5-x^6-2 x^8+2 x^{10}} \, dx+5 \int \frac {x^8 \sqrt [3]{1+x^3+x^5}}{2-2 x^3+4 x^5-x^6-2 x^8+2 x^{10}} \, dx\\ \end {align*}

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Mathematica [F]  time = 0.86, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (1+x^5\right ) \sqrt [3]{1+x^3+x^5} \left (-3+2 x^5\right )}{x^2 \left (2-2 x^3+4 x^5-x^6-2 x^8+2 x^{10}\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Integrate[((1 + x^5)*(1 + x^3 + x^5)^(1/3)*(-3 + 2*x^5))/(x^2*(2 - 2*x^3 + 4*x^5 - x^6 - 2*x^8 + 2*x^10)),x]

[Out]

Integrate[((1 + x^5)*(1 + x^3 + x^5)^(1/3)*(-3 + 2*x^5))/(x^2*(2 - 2*x^3 + 4*x^5 - x^6 - 2*x^8 + 2*x^10)), x]

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IntegrateAlgebraic [A]  time = 2.77, size = 116, normalized size = 1.00 \begin {gather*} \frac {3 \sqrt [3]{1+x^3+x^5}}{2 x}-\frac {1}{4} \text {RootSum}\left [3-6 \text {$\#$1}^3+2 \text {$\#$1}^6\&,\frac {-3 \log (x)+3 \log \left (\sqrt [3]{1+x^3+x^5}-x \text {$\#$1}\right )+4 \log (x) \text {$\#$1}^3-4 \log \left (\sqrt [3]{1+x^3+x^5}-x \text {$\#$1}\right ) \text {$\#$1}^3}{-3 \text {$\#$1}^2+2 \text {$\#$1}^5}\&\right ] \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[((1 + x^5)*(1 + x^3 + x^5)^(1/3)*(-3 + 2*x^5))/(x^2*(2 - 2*x^3 + 4*x^5 - x^6 - 2*x^8 + 2*x^
10)),x]

[Out]

(3*(1 + x^3 + x^5)^(1/3))/(2*x) - RootSum[3 - 6*#1^3 + 2*#1^6 & , (-3*Log[x] + 3*Log[(1 + x^3 + x^5)^(1/3) - x
*#1] + 4*Log[x]*#1^3 - 4*Log[(1 + x^3 + x^5)^(1/3) - x*#1]*#1^3)/(-3*#1^2 + 2*#1^5) & ]/4

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fricas [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^5+1)*(x^5+x^3+1)^(1/3)*(2*x^5-3)/x^2/(2*x^10-2*x^8-x^6+4*x^5-2*x^3+2),x, algorithm="fricas")

[Out]

Exception raised: TypeError >>  Error detected within library code:   integrate: implementation incomplete (tr
ace 0)

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {{\left (2 \, x^{5} - 3\right )} {\left (x^{5} + x^{3} + 1\right )}^{\frac {1}{3}} {\left (x^{5} + 1\right )}}{{\left (2 \, x^{10} - 2 \, x^{8} - x^{6} + 4 \, x^{5} - 2 \, x^{3} + 2\right )} x^{2}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^5+1)*(x^5+x^3+1)^(1/3)*(2*x^5-3)/x^2/(2*x^10-2*x^8-x^6+4*x^5-2*x^3+2),x, algorithm="giac")

[Out]

integrate((2*x^5 - 3)*(x^5 + x^3 + 1)^(1/3)*(x^5 + 1)/((2*x^10 - 2*x^8 - x^6 + 4*x^5 - 2*x^3 + 2)*x^2), x)

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maple [F]  time = 180.00, size = 0, normalized size = 0.00 \[\int \frac {\left (x^{5}+1\right ) \left (x^{5}+x^{3}+1\right )^{\frac {1}{3}} \left (2 x^{5}-3\right )}{x^{2} \left (2 x^{10}-2 x^{8}-x^{6}+4 x^{5}-2 x^{3}+2\right )}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x^5+1)*(x^5+x^3+1)^(1/3)*(2*x^5-3)/x^2/(2*x^10-2*x^8-x^6+4*x^5-2*x^3+2),x)

[Out]

int((x^5+1)*(x^5+x^3+1)^(1/3)*(2*x^5-3)/x^2/(2*x^10-2*x^8-x^6+4*x^5-2*x^3+2),x)

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {{\left (2 \, x^{5} - 3\right )} {\left (x^{5} + x^{3} + 1\right )}^{\frac {1}{3}} {\left (x^{5} + 1\right )}}{{\left (2 \, x^{10} - 2 \, x^{8} - x^{6} + 4 \, x^{5} - 2 \, x^{3} + 2\right )} x^{2}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^5+1)*(x^5+x^3+1)^(1/3)*(2*x^5-3)/x^2/(2*x^10-2*x^8-x^6+4*x^5-2*x^3+2),x, algorithm="maxima")

[Out]

integrate((2*x^5 - 3)*(x^5 + x^3 + 1)^(1/3)*(x^5 + 1)/((2*x^10 - 2*x^8 - x^6 + 4*x^5 - 2*x^3 + 2)*x^2), x)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int -\frac {\left (x^5+1\right )\,\left (2\,x^5-3\right )\,{\left (x^5+x^3+1\right )}^{1/3}}{x^2\,\left (-2\,x^{10}+2\,x^8+x^6-4\,x^5+2\,x^3-2\right )} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-((x^5 + 1)*(2*x^5 - 3)*(x^3 + x^5 + 1)^(1/3))/(x^2*(2*x^3 - 4*x^5 + x^6 + 2*x^8 - 2*x^10 - 2)),x)

[Out]

int(-((x^5 + 1)*(2*x^5 - 3)*(x^3 + x^5 + 1)^(1/3))/(x^2*(2*x^3 - 4*x^5 + x^6 + 2*x^8 - 2*x^10 - 2)), x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (x + 1\right ) \left (2 x^{5} - 3\right ) \sqrt [3]{x^{5} + x^{3} + 1} \left (x^{4} - x^{3} + x^{2} - x + 1\right )}{x^{2} \left (2 x^{10} - 2 x^{8} - x^{6} + 4 x^{5} - 2 x^{3} + 2\right )}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x**5+1)*(x**5+x**3+1)**(1/3)*(2*x**5-3)/x**2/(2*x**10-2*x**8-x**6+4*x**5-2*x**3+2),x)

[Out]

Integral((x + 1)*(2*x**5 - 3)*(x**5 + x**3 + 1)**(1/3)*(x**4 - x**3 + x**2 - x + 1)/(x**2*(2*x**10 - 2*x**8 -
x**6 + 4*x**5 - 2*x**3 + 2)), x)

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